Mathematical Foundation of AI Research Centre

Book here

Zoom

Semi-supervised learning is the problem of finding missing labels; more precisely one has a data set of feature vectors of which a (often small) subset are labelled. The semi-supervised learning assumption is that similar feature vectors should have similar labels which implies one needs a geometry on the set of feature vectors. A typical way to represent this geometry is via a graph where the nodes are the feature vectors and the edges are weighted by some measure of similarity. Laplace learning is a popular graph-based method for solving the semi-supervised learning problem which essentially requires one to minimise a Dirichlet energy defined on the graph (hence the Euler-Lagrange equation is Laplace's equation). However, at low labelling rates Laplace learning typically performs poorly. This is due to the lack of regularity, or the ill-posedness, of solutions to Laplace's equation in any dimension higher (or equal to) two. The random walk interpretation allows one to characterise how close one is to entering the ill-posed regime. In particular, it allows one to give a lower bound on the number of labels required and even provides a route for correcting the bias. Correcting the bias leads to a new method, called Poisson learning. Finally, the ideas behind correcting the bias in Laplace learning have motivated a new graph neural network architecture which do not suffer from the over-smoothing phenomena. In particular, this type of neural network, which we call GRAND++ (GRAph Neural Diffusion with a source term) enables one to employ deep architectures. 

Connectivity and percolation are two well studied phenomena in random graphs. 

In this talk we will discuss higher-dimensional analogues of connectivity and percolation that occur in random simplicial complexes.

Simplicial complexes are a natural generalization of graphs that consist of vertices, edges, triangles, tetrahedra, and higher dimensional simplexes.

We will mainly focus on random geometric complexes. These complexes are generated by taking the vertices to be a random point process, and adding simplexes according to their geometric configuration.

Our generalized notions of connectivity and percolation use the language of homology - an algebraic-topological structure representing cycles of different dimensions.

In this talk we will discuss recent results analyzing phase transitions related to these topological phenomena. 

Zoom link

Graph signal processing (GSP) tries to device appropriate tools to process signals supported on graphs by generalizing classical methods from signal processing of time-series and images -- such as smoothing, filtering and supported on the nodes of graphs.  Typically, this involves leveraging the structure of the graph as encoded in the spectral properties of the graph Laplacian. In applications such as traffic network analysis, however, the signals of interest are naturally defined on the edges of a graph, rather than on the nodes. After a very brief recap of the central ideas of GSP, we examine why the standard tools from GSP may not be suitable for the analysis of such edge signals.  More specifically, we discuss how the underlying notion of a 'smooth signal' inherited from (the typically considered variants of) the graph Laplacian are not suitable when dealing with edge signals that encode flows.  To overcome this limitation we devise signal processing tools based on the Hodge-Laplacian and the associated discrete Hodge Theory for simplicial (and cellular) complexes.  We discuss applications of these ideas for signal smoothing, semi-supervised and active learning for edge-flows on discrete (or discretized) spaces.

Zoom link

In this talk, we explore some of the benefits that the mathematical theories of convex optimisation and regularisation can bring to selected machine learning applications. We begin with a discussion of the well-known  least absolute shrinkage and selection operator (LASSO, Tibshirani 1996) and explore how to determine if a desired solution is in the range of this operator. From there, we move on to the training of perceptrons and deep neural networks. In modern literature, a combination of gradient-based optimisation method and gradient computation via backpropagation is often the method of choice when it comes to training deep neural networks. In this talk, we focus on alternative training strategies that are based on recent developments in distributed optimisation of deeply nested systems (Carreira-Perpinan & Wang 2014). In particular, we propose a novel training approach that is based on proximal activation functions and generalised Bregman distances. The main advantages of this approach compared to previous approaches is that it enables (potentially distributed) training of neural network parameters without having to differentiate activation functions. This work is joint work with Xiaoyu Wang from the University of Cambridge.

Zoom link

The talk will also be live streamed on zoom. To join, please use the following link: https://qmul-ac-uk.zoom.us/j/5928131069

The iterated conditional Sequential Monte Carlo (cSMC) method is a particle MCMC method commonly used for state inference in non-linear, non-Gaussian state space models. Standard implementations of iterated cSMC provide an efficient way to sample state sequences in low-dimensional state space models. However, efficiently scaling iterated cSMC methods to perform well in models with a high-dimensional state remains a challenge. One reason for this is the use of a global proposal, without reference to the current state sequence. In high dimensions, such a proposal will typically not be well-matched to the posterior and impede efficient sampling. I will describe techniques based on the embedded HMM (Hidden Markov Model) framework to construct efficient proposals in high dimensions that are local relative to the current state sequence.

Abstract: Every time a cell divides, it must copy (or “replicate”) its genome exactly once, which it achieves through the parallel action of thousands of replication forks. One of the most serious errors in DNA replication occurs when replication forks stall, which happens when the fork encounters an obstacle that it cannot pass. The frequent slowing or stalling of replication forks, termed “replication stress”, is rare in healthy human cells but common in both cancer cells and parasites. Replication stress is therefore a common therapeutic target for anti-malarial and cancer chemotherapies, but we have a relatively poor understanding of where, when, why, and how often replication forks stall under these therapies. I will discuss our recent progress towards answering these questions, whereby we are using long-read nanopore DNA sequencing together with AI to measure the movement and stress of thousands of replication forks across the genomes of human cancer cells and the malaria parasite Plasmodium falciparum. 

I will describe a geometric approach to the problem of constructing algorithms for autonomous robot motion.

Zoom link: https://qmul-ac-uk.zoom.us/j/5928131069

Join Zoom Meeting
https://qmul-ac-uk.zoom.us/j/5928131069